As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed:
where $P_l^m(\cos\theta)$ are the Associated Legendre Polynomials.
I believe that the answer is ultimately 0, but I am not quite sure of how reduce this integrable to a workable form, I have tried using recursion relations but have had no luck. I am sure that I am not trying to solve the integrand with an additional $\sin(\theta)$ factor, which would be easily solved using the change of variable $z = \cos(\theta)$ and applying the orthogonality of the Associated Legendre Polynomials. Does anyone have any advice?